Abstract
We establish necessary and sufficient conditions under which a real-valued function from \(L_p (\mathbb{T})\), 1 ≤ p < ∞, is badly approximable by the Hardy subspace H 0p := {ƒ ∈ H p : ƒ(0) = 0}. In a number of cases, we obtain the exact values of the best approximations in the mean of functions holomorphic in the unit disk by functions holomorphic outside this disk. We use the obtained results for finding the exact values of the best polynomial approximations and n-widths of some classes of holomorphic functions. We establish necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1047–1067, August, 2007.
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Savchuk, V.V. Best approximation by holomorphic functions. Application to the best polynomial approximation of classes of holomorphic functions. Ukr Math J 59, 1163–1183 (2007). https://doi.org/10.1007/s11253-007-0078-0
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DOI: https://doi.org/10.1007/s11253-007-0078-0